Skip to main content

Stable probability measures on groups and on vector spaces

A survey

  • Survey Articles
  • Conference paper
  • First Online:
Probability Measures on Groups VIII

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1210))

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Araujo, E. Giné: The central limit theorem for real and Banach valued random variables. J. Wiley, New York (1980).

    MATH  Google Scholar 

  2. P. Baldi: Lois stables sur les déplacements de ℝd. In: Probability measures on groups. Proceedings Oberwolfach (1978). Lecture Notes in Math. 706, 1–9. Springer (1979).

    Google Scholar 

  3. N.H. Bingham: Factorization theory and domains of attraction for generalized convolution algebras. Proc. London Math. Soc. (3) 23, 16–30 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  4. H. Carnal: Les variables aléatoires de loi stable et leur représentation selon P. Lévy. In: Probability measures on groups VIII. Proceedings Oberwolfach (1985). Lecture Notes Math. Springer (1986).

    Google Scholar 

  5. P. Crépel: Grenzwetsätze für abhängige Zufallsvariable und Irrfahrten auf Gruppen. In: Probability measures on groups. Proceedings Oberwolfach (1978). Lecture Notes Math. 706, 54–66. Springer (1979).

    Google Scholar 

  6. E. Dettweiler: Stabile Maße auf Badrikianschen Räumen. Math. Z. 146, 149–166 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  7. T. Drisch, L. Gallardo: Stable laws on the Heisenberg group. In: Probability measures on groups. Proccedings Oberwolfach (1983). Lecture Notes Math. 1064, 56–79 (1984)

    Google Scholar 

  8. T. Drisch, L. Gallardo: Stable laws on the diamond group. In preparation.

    Google Scholar 

  9. L. Gallardo: Processuss subordonnés au mouvement brownien sur les groupes de Lie nilpotents. Compt. Rend. Acad. Sc. Paris 292, 413–416 (1981).

    MathSciNet  MATH  Google Scholar 

  10. L. Gallardo: Processus subordonnés et mouvement brownien sur les groupes de Lie nilpotents. In: Marches aléatoires et processus stochastiques sur les groupes de Lie, Nancy (1981) Inst. E. Cartan 40–52 (1983).

    Google Scholar 

  11. Y. Guivarc'h, M. Keane, B. Roynette: Marches aléatoires sur les groupes de Lie. Lecture Notes Math. 624, Springer (1977).

    Google Scholar 

  12. G. Forst: A characterization of self-decomposable probabilities on the half-line. Z. Wahrscheinlichkeitstheorie verw. Geb. 49, 349–352 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Głowacki: A calculus of symbols and convolution semigroups on the Heisenberg group. Studia Math. 72, 291–321 (1982).

    MathSciNet  MATH  Google Scholar 

  14. P. Głowacki: Stable semigroups of measures on the Heisenberg group. Studia Math.

    Google Scholar 

  15. P. Głowacki, A. Hulanicki: A semigroup of probability measures with non-smooth differentiable densities on a Lie group. Preprint (1984).

    Google Scholar 

  16. P. Głowacki: On commutative approximate identities on non-graded homogeneous groups. Comm. Part. Differential Equ. 9, 979–1016 (1984).

    Article  MATH  Google Scholar 

  17. P. Głowacki: Stable semigroups of measures as commutative approximate identities on non-graded homogeneous groups. Preprint (1985).

    Google Scholar 

  18. U. Grenander: Probabilities on algebraic structures. Almquist & Wiksell, Upsala (1963).

    MATH  Google Scholar 

  19. M.G. Hahn, M.J. Klass: A survey of generalized domains of attraction and operator norming methods. In: Probability in Banach spaces III, Proceedings Medford (1980). Lecture Notes Math. 860, 187–218 (1981).

    Google Scholar 

  20. M.G. Hahn, M.J. Klass: Affine normality of partial sums of i.i.d. random vectors; A characterization. Z. Wahrscheinlichkeitstheorie verw. Geb. 68, 479–505 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  21. W. Hazod: Subordination von Faltungs-und Operatorhalbgruppen. In: Probability measures on groups. Proceedings Oberwolfach (1978). Lecture Notes Math. 706, 144–202 (1979).

    Google Scholar 

  22. W. Hazod: Stable probabilities on locally compact groups. In: Probability measures on groups. Proceedings Oberwolfach (1981). Lecture Notes Math. 928, 183–211 (1982).

    Google Scholar 

  23. W. Hazod: Remarks on [semi-] stable probabilities. In: Probability measures on groups. Proceedings Oberwolfach (1983). Lecture Notes Math. 1064, 182–203 (1984).

    Google Scholar 

  24. W. Hazod: Stable and semistable probabilities on groups and vector spaces. In: Probability theory on vector spaces III. Proceedings Lublin (1983). Lecture Notes Math. 1080, 69–89 (1984).

    Google Scholar 

  25. W. Hazod: Semigroupes de convolution [demi-] stables et autodécomposables sur les groupes localement compacts. In: Probabilités sur les structures géometriques. Actes des Journees Toulouse (1984). Publ. du Lab. Stat. et Prob. Université de Toulouse, 57–85 (1985).

    Google Scholar 

  26. W. Hazod, E. Siebert: Continuous automorphism groups on a locally compact group contracting modulo a compact subgroup and applications to stable convolution semigroups. Semigroup Forum (1986). To appear.

    Google Scholar 

  27. H. Heyer: Probability measures on locally compact groups. Ergebnisse der Math. Berlin-Heidelberg-New York. Springer (1977).

    Google Scholar 

  28. H. Heyer: Probability theory on hypergroups: A survey. In: Probability measures on Groups VII. Proceedings Oberwolfach (1983). Lecture Notes Math. 1064, 481–550 (1984).

    Google Scholar 

  29. K.H. Hofmann, P. Mostert: Splitting in topological groups. Mem. Amer. Math. Soc. 43 (1963).

    Google Scholar 

  30. J.P. Holmes, W.N. Hudson, J.D. Mason: Operator-stable laws: multiple exponents and elliptical symmetry. Ann. Probab. 10, 602–612 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  31. W.N. Hudson: Operator-stable distributions and stable marginals. J. Mult. Analysis 10, 26–37 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  32. W.N. Hudson, J.D. Mason: Exponents of operator stable laws. In: Probability in Banach spaces III. Proceedings Medford (1980). Lecture Notes in Math. 860, 291–298 (1981).

    Google Scholar 

  33. W.N. Hudson, J.D. Mason: Operator stable laws. J. Mult. Analysis 11, 434–447 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  34. W.N. Hudson, J.D. Mason: Operator stable measures on ℝ2 with multiple exponents. Ann. Prob. 9, 482–489 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  35. W.N. Hudson, J.D. Mason, J.A. Veeh: The domain of normal attraction of an operator-stable law. Ann. Prob. 11, 178–184 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  36. W.N. Hudson, J.D. Mason: Operator-self-similar processes in a finite dimensional space. Trans. Am. Math. Soc. 273, 281–297 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Hulanicki: The distribution of energy in the Brownian motion in the gaussian field and analytic hypoellipticity of certain subelliptic operators on the Heisenberg group. Studia Math. 16, 165–173 (1976).

    MathSciNet  MATH  Google Scholar 

  38. A. Hulanicki: A Tauberian property of the convolution semigroup generated by X2-|Y|r on the Heisenberg group. Proceedings Symposia Pure Math. (AMS) 35, 403–405 (1979).

    Article  MathSciNet  Google Scholar 

  39. A. Hulanicki: A class of convolution semi-groups of measures on a Lie group. In: Probability theory on vector spaces II. Proceedings Błzźejewko (1979). Lecture Notes Math. 828, 82–101 (1980).

    Google Scholar 

  40. R. Jajte: Quasi stable measures in generalized convolution algebras. Bull. Acad. Sci. Pol. 24, 505–511 (1976).

    MathSciNet  MATH  Google Scholar 

  41. R. Jajte: Quasi stable measures in generalized convolution algebras II. Bull. Acad. Sci. Pol. 25, 67–72 (1977).

    MathSciNet  MATH  Google Scholar 

  42. R. Jajte: Semistable probability measures on ℝN. Studia Math. 61, 29–39 (1977).

    MathSciNet  MATH  Google Scholar 

  43. R. Jajte: V-decomposable measures on a Hilbert space. In: Probability theory on vector spaces II. Proceedings Błaźejewko (1979). Lecture Notes in Math. 828, 108–127 (1980).

    Google Scholar 

  44. A. Janssen, H. Milbrodt, H. Strasser: Infinitely divisible statistical expeiments. Lecture Notes in Statistics 27 (1985).

    Google Scholar 

  45. A. Janssen: A survey about zero — one — laws for probability measures on linear spaces and locally compact groups. In: Probability measures on groups VII. Proceedings Oberwolfach (1983). Lecture Notes Math. 1064, 551 — 563 (1984).

    Google Scholar 

  46. A. Janssen: Unendlich teilbare statistische Experimente. Habilitationsschrift, Universität Dortmund (1982).

    Google Scholar 

  47. Z.J. Jurek: A limit theorem for truncated random variables. Bull. Acad. Pol. Sci. 23, 911–913 (1975).

    MathSciNet  MATH  Google Scholar 

  48. Z.J. Jurek: Limit distributions for shrunken random variables. Diss. Math. 85, 1–46 (1981).

    MathSciNet  MATH  Google Scholar 

  49. Z.J. Jurek: On Gaussian measures on IRd. In: Proceedings of the Sixth Conference on Probability Theory, Braşov (1979). Ed. Acad. Rep. S. Romania, Buckarest (1981).

    Google Scholar 

  50. Z.J. Jurek: On stability of probability measures in Euclidean spaces. In: Probability theory on vector spaces II. Proceedings Błażejewko (1979). Lecture Notes Math. 828, 129–145 (1980).

    Google Scholar 

  51. Z.J. Jurek: Central limit theorem in Euclidean spaces. Bull. Acad. Pol. Sci. Math. 28, 81–86 (1980).

    MathSciNet  MATH  Google Scholar 

  52. Z.J. Jurek: Limit distributions and one parameter groups of linear operators on Banach spaces. J. Multiv. Anal. 13, 578–604 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  53. Z. Jurek: Domains of normal attraction of operator — stable measures on Euclidean spaces. Bull. Acad. Pol. Sci., 28, 397–406 (1980).

    MathSciNet  MATH  Google Scholar 

  54. Z.J. Jurek: Remarks on V-decomposable measures. Bull. Acad. Pol. Sci. 30, 393–401 (1982).

    MathSciNet  MATH  Google Scholar 

  55. Z.J. Jurek: Domains of normal attraction for G-stable measures on IRd. Theory Prob. Appl. 27, 396–400 (1982).

    MathSciNet  MATH  Google Scholar 

  56. Z.J. Jurek: Convergence of types, self-decomposibility and stability of measures on linear spaces. In: Probability in Banach spaces III. Proceedings Medford (1980). Lecture Notes in Math. 860, 257–284 (1981).

    Google Scholar 

  57. Z.J. Jurek: Polar coordinates in Banach spaces. Bull. Acad. Polon. Sci. Math. 32, 61–66 (1984).

    MathSciNet  MATH  Google Scholar 

  58. Z.J. Jurek: Random integral representations for classes of limit distributions similar to Lévy class L*o. Preprint (1985).

    Google Scholar 

  59. E. Kehrer: Stabilität von Wahrscheinlichkeitsmaßen unter Operatorgruppen auf Banachräumen. Dissertation, Universität Tübingen (1983).

    Google Scholar 

  60. Yu.S. Khokhlow: On the convergence to a multi-dimensional stable law of the distribution of a shift parameter for the composition of random motions in Euclidean space. Theory Prob. Appl. 27, 363–365 (1982).

    Article  Google Scholar 

  61. S. Wah Kiu: Semistable Markov processes in IRn. Stoch. Proc. Appl. 10, 183–191 (1980).

    Article  MATH  Google Scholar 

  62. W. Krakowiak: Operator stable probability measures on Banach spaces. Coll. Math. 41, 313–326 (1979).

    MathSciNet  MATH  Google Scholar 

  63. J. Kucharczak: On operator stable probability measures. Bull. Acad. Pol. Sci. 23, 571–576 (1975).

    MathSciNet  MATH  Google Scholar 

  64. J. Kucharczak: Remarks on operator stable measures. Coll. Math. 34, 109–119 (1976).

    MathSciNet  MATH  Google Scholar 

  65. J. Kucharczak, K. Urbanik: Operator stable probability measures on some Banach space. Bull. Acad. Pol. Sci. 25, 585–588 (1977).

    MathSciNet  MATH  Google Scholar 

  66. R.G. Laha, V.K. Rohatgi: Self-similar stochastic processes in a Hilbert space. Preprint (1982).

    Google Scholar 

  67. R.G. Laha, V.K. Rohatgi: Operator self-similar stochastic processes in IRd. Stoch. Proc. Appl. 12, 73–84 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  68. J. Lamperti: Semistable stochastic processes. Trans. Amer. Math. Soc. 104, 62–78 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  69. R. LePage, M. Woodruffe, J. Zinn: Convergence to a stable distribution via order statistics. Ann. Prob. 9, 624–632 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  70. R. LePage: Multidimensional infinitely divisible variables and processes. Part II. In: Probability in Banach spaces III. Proceedings Medford (1980). Lecture Notes Math. 860, 279–284 (1981).

    Google Scholar 

  71. P. Lévy: Propriétés asymptotiques des sommes de variables aléatoires indépendantes ou enchaînées. Journal de Math. 14, fasc. IV. (1935).

    Google Scholar 

  72. W. Linde: Infinitely divisible and stable measures on Banach spaces. Teubner Texte zur Math. vol. 58, Leipzig (1983).

    Google Scholar 

  73. A. Łuczak: Operator semi-stable probability measures on IRN. Coll. Math. 45, 287–299 (1981).

    MATH  Google Scholar 

  74. A. Łuczak: Elliptical symmetry and characterization of operator-stable and operator semi-stable measures. Ann. Probab. 12, 1217–1223 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  75. A. Łuczak: Independent marginals of a probability measure. Preprint (1983).

    Google Scholar 

  76. M.B. Marcus, G. Pisier: Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes. Acta Math. 152, 245–301 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  77. J. Michaliček: Der Anziehungsbereich von operatorstabilen Verteilungen im IR2. Z. Wahrscheinlichkeitstheorie verw. Geb. 25, 57–70 (1972).

    Article  MATH  Google Scholar 

  78. J. Michaliček: Die Randverteilungen der operatorstabilen Maße im zweidimensionalen Raum. Z. Wahrscheinlichkeitstheorie verw. Geb. 21, 135–146 (1972).

    Article  MATH  Google Scholar 

  79. B. Mincer, K. Urbanik: Completely stable measures on Hilbert spaces. Coll. Math. 42, 301–307 (1979).

    MathSciNet  MATH  Google Scholar 

  80. K.R. Parthasarathy: Every completely stable distribution is normal. Sankhya 35, Ser. A, 35–38 (1973).

    MathSciNet  MATH  Google Scholar 

  81. K.R. Parthasarathy, K. Schmidt: Stable positive definite functions. Trans. Amer. Math. Soc. 203, 161–174 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  82. U. Pickartz: Semiflüsse auf Räumen von Wahrscheinlichkeitsmaßen. Dissertation, Universität Dortmund (1983).

    Google Scholar 

  83. A. Raugi: Théorème de limite centrale pour un produit semidirect d'un groupe de Lie résoluble simplement connexe de type rigide par un groupe compact. In: Probability measures on groups. Proceedings Oberwolfach (1978). Lecture Notes Math. 706, 257–324 (1979).

    Google Scholar 

  84. E.L. Rvačeva: Domains of attraction of multidimensional distributions. Select. Transl. Math. Stat. Prob. 2, 183–205 (1962).

    Google Scholar 

  85. K. Schmidt: Stable probability measures on ℝ. Z. Wahrscheinlichkeitstheorie verw. Geb. 33, 19–31 (1975).

    Article  MathSciNet  Google Scholar 

  86. S.V. Semovskii: Operator stable laws of distributions. Sovjet Math. Doklady 20, 139–142 (1979).

    MathSciNet  Google Scholar 

  87. S.V. Semovskii: The central limit theorem for sums of random vectors normalized by linear operators. Sovjet Math. Doklady 20, 356–359 (1979).

    MathSciNet  Google Scholar 

  88. M. Sharpe: Operator stable probability measures on vector groups. Trans. Amer. Math. Soc. 136, 51–65 (1969).

    Article  MathSciNet  MATH  Google Scholar 

  89. E. Siebert: Fourier analysis and limit theorems for convolution semigroups on a locally compact group. Adv. Math. 39, 111–154 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  90. E. Siebert: Semistable convolution semigroups on measurable and topological groups. Ann. Inst. H. Poincaré 20, 147–164 (1984).

    MathSciNet  MATH  Google Scholar 

  91. E. Siebert: Supplements to operator-stable and operator-semistable laws on Euclidean spaces. To appear in J. Multivariate Anal.

    Google Scholar 

  92. E. Siebert: Contractive autmorphisms on locally compact groups. Math. Z. 191, 73–90 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  93. E. Siebert: Holomorphic convolution semigroups on topological groups. In: Probability measures on groups VII. Proceedings Oberwolfach (1983). Lecture Notes Math. 1064, 421–449 (1984).

    Google Scholar 

  94. E. Stein, A. Nagel: Lectures on pseudodifferential-operators. Math. Notes. Princeton Univ. Press (1979).

    Google Scholar 

  95. E. Stein, G.B. Folland: Hardy spaces on homogeneous groups. Math. Notes, Princeton Univ. Press (1982).

    Google Scholar 

  96. F.W. Steutel, K. van Harn: Discrete analogues of self-decomposability and stability. Ann. Probability 7, 893–899 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  97. H. Strasser: Scale invariance of statistical experiments. Probability and Math. Stat. 5, 1–20 (1985).

    MathSciNet  MATH  Google Scholar 

  98. R. Sztencel: On the lower tail of stable seminorm. Bull. Pol. Acad. Sci. Math. 32, 11–12 (1984).

    MathSciNet  MATH  Google Scholar 

  99. R. Sztencel: Absolute continuity of the lower tail of stable seminorm. Preprint (1985).

    Google Scholar 

  100. A. Tortrat: Lois stables dans un groupe. Ann. Inst. H. Poincaré, Sect. B. 17, 51–61 (1981).

    MathSciNet  MATH  Google Scholar 

  101. A. Tortrat: Lois zero-un et lois semi-stables dans un groupe. In: Probability measures on groups. Proceedings Oberwolfach (1981). Lecture Notes in Math. 928, 452–466 (1982).

    Google Scholar 

  102. K. Urbanik: Lévy's probability measures on Euclidean space. Studia Math. 44, 119–148 (1972).

    MathSciNet  MATH  Google Scholar 

  103. K. Urbanik: Lévy's probability measures on Banach spaces. Studia Math. 63, 283–308 (1978).

    MathSciNet  MATH  Google Scholar 

  104. K. Urbanik: Generalized convolutions. Studia Math. 23, 217–245 (1964).

    MathSciNet  MATH  Google Scholar 

  105. K. Urbanik: Generalized convolutions II. Studia Math. 45, 57–70 (1973).

    MathSciNet  MATH  Google Scholar 

  106. K. Urbanik: Generalized convolutions III. Studia Math. 80, 167–189 (1984).

    MathSciNet  MATH  Google Scholar 

  107. K. Urbanik: Generalized convolutions IV. Preprint (1984).

    Google Scholar 

  108. P. Vatan: Max-infinite divisibility and max-stability in infinite dimensions. In: Probability in Banach spaces V. Proceedings, Medford (1984). Lecture Notes Math. 1153, 400–425 (1985).

    Google Scholar 

  109. J.A. Veeh: Infinitely divisible measures with independent marginals. Z. Wahrscheinlichkeitstheorie verw. Geb. 61, 303–308 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  110. A. Weron: Stable processes and measures: A survey. In: Probability theory on vector spaces III. Proceedings Lublin (1983). Lecture Notes Math. 1080, 307–364 (1984).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Herbert Heyer

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Springer-Verlag

About this paper

Cite this paper

Hazod, W. (1986). Stable probability measures on groups and on vector spaces. In: Heyer, H. (eds) Probability Measures on Groups VIII. Lecture Notes in Mathematics, vol 1210. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077190

Download citation

  • DOI: https://doi.org/10.1007/BFb0077190

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16806-5

  • Online ISBN: 978-3-540-44852-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics